Lecture12 Rosalind Archer

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Presentation transcript:

Lecture12 Rosalind Archer PETE 324 Lecture12 Rosalind Archer

Real Gas Diffusivity Equation The real gas diffusivity equation can be derived in a similar manner to the (slightly compressible) liquid case.

Real Gas Diffusivity Equation Starting by substituting the equation of state and Darcy’s Law into the continuity equation: Cancel the constants M, R and T from each side.

Real Gas Diffusivity Equation Assume that the permeability, k, is a constant:

Real Gas Diffusivity Equation Now use the chain rule to expand the time derivative terms. The right hand side of the equation becomes:

Real Gas Diffusivity Equation Recalling the definition of the rock compressibility: Also recall the definition of isothermal gas compressibility:

Real Gas Diffusivity Equation Therefore:

Real Gas Diffusivity Equation Substituting the definitions of compressibility into the right hand side of the diffusivity equation gives:

Real Gas Diffusivity Equation The final form of the real gas diffusivity equation is:

Real Gas Diffusivity Equation The equation is nonlinear because the terms multiplying (p, z, m) are all functions of the unknown p. There are also nonlinearities on the right hand side as well.

Pseudopressure The equation can be linearised by introducing a pseudo-pressure:

Pseudopressure With the introduction of the gas pseudopressure the diffusivity equation becomes:

Pseudopressure The derivative of the gas pseudopressure with respect to pressure is: Substituting this into the diffusivity equation gives:

Pseudopressure Cancelling like terms gives: Use of pseudopressure has linearised the right hand side of the equation but the left hand side still has some pressure depend terms (m,ct).

Pseudotime A pseudotime formulation to linearise the left hand side was proposed by Agarwal: Using pseudopressure and pseudotime gives: